# cell attachment

Let $X$ be a topological space, and let $Y$ be the adjunction $Y:=X\cup_{\varphi}D^{k}$, where $D^{k}$ is a closed $k$-ball (http://planetmath.org/StandardNBall) and $\varphi\colon S^{k-1}\rightarrow X$ is a continuous map, with $S^{k-1}$ is the $(k-1)$-sphere considered as the boundary of $D^{k}$. Then, we say that $Y$ is obtained from $X$ by the attachment of a $k$-cell, by the attaching map $\varphi.$ The image $e^{k}$ of $D^{k}$ in $Y$ is called a closed $k$-cell, and the image $\smash{\overset{\circ}{e}}^{k}$ of the interior

 $D^{\circ}:=D^{k}\setminus S^{k-1}$

of $D^{k}$ is the corresponding open $k$-cell.

Note that for $k=0$ the above definition reduces to the statement that $Y$ is the disjoint union of $X$ with a one-point space.

More generally, we say that $Y$ is obtained from $X$ by cell attachment if $Y$ is homeomorphic to an adjunction $X\cup_{\left\{\varphi_{i}\right\}}D^{k_{i}}$, where the maps ${\left\{\varphi_{i}\right\}}$ into $X$ are defined on the boundary spheres of closed balls ${\left\{D^{k_{i}}\right\}}$.

 Title cell attachment Canonical name CellAttachment Date of creation 2013-03-22 13:25:53 Last modified on 2013-03-22 13:25:53 Owner yark (2760) Last modified by yark (2760) Numerical id 13 Author yark (2760) Entry type Definition Classification msc 54B15 Synonym cell adjunction Related topic CWComplex Defines cell Defines open cell Defines closed cell Defines attaching map